Neutron lens by superposition of glancing reflectionsA. D. Stoica and X.-L. Wang Citation: Review of Scientific Instruments 74, 2463 (2003); doi: 10.1063/1.1556946 View online: http://dx.doi.org/10.1063/1.1556946 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/74/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Application of the rigorous method to x-ray and neutron beam scattering on rough surfaces J. Appl. Phys. 108, 033516 (2010); 10.1063/1.3467937 A monolithic polycapillary focusing optic for polychromatic neutron diffraction applications Rev. Sci. Instrum. 73, 1985 (2002); 10.1063/1.1470236 Molecular lens applied to benzene and carbon disulfide molecular beams J. Chem. Phys. 114, 8293 (2001); 10.1063/1.1367380 Cold neutron microprobe for materials analysis using tapered capillary optics Rev. Sci. Instrum. 71, 3247 (2000); 10.1063/1.1288262 A polycapillary bending and focusing lens for neutrons Rev. Sci. Instrum. 68, 3744 (1997); 10.1063/1.1148020

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Neutron lens by superposition of glancing reflectionsA. D. Stoicaa) and X.-L. Wangb)

Spallation Neutron Source, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830

~Received 15 August 2002; accepted 25 November 2002!

A neutron lens can lead to substantial intensity gains in neutron scattering experiments by imaginga source directly onto the sample. A design is proposed for a compact neutron lens made of a stackof silicon wafers, each coated with a neutron-reflective supermirror on one side and aneutron-absorbing layer on the other. We show that for imaging, the length of each mirror is afunction of its distance from the optical axis,y, following a simple 1/y relationship. Uniformbending of the mirror assembly decreases the spatial aberration. The optimal design to minimize theoptical spatial aberrations is discussed and Monte Carlo simulation results are presented. ©2003American Institute of Physics.@DOI: 10.1063/1.1556946#

Despite the advances in neutron source and instrumenta-tion, many neutron scattering experiments remain flux lim-ited. Considerable efforts are underway to explore ways offocusing a neutron beam, or imaging a neutron source, bytotal external reflection.1–3 The single mirror approach em-ploys parabolic or elliptic reflecting surfaces and has workedwell for imaging x-ray sources.4 However, the field of viewachieved with a single mirror is quite limited as significantaberrations occur for a point source located off the opticalaxis ~defined as the line connecting the foci!. In addition,perfectly shaped mirrors demand a high precision in machin-ing and are extremely intolerant to misalignment, especiallyfor glancing incidence. Polycapillary optics has shown excit-ing capabilities to focus neutrons at submillimeter scale.2,3

Unfortunately, the devices designed on this principle havelow transmission, large angular divergence at focus~;10°!,and a rather short focusing distance~,10 cm!. These fea-tures have limited the applications of polycapillary lens inneutron scattering.

The idea of imaging by superposition of glancing reflec-tions was first proposed by Schmidt in 1975.5 In the sameyear, Vogt reported the discovery of a natural prototype, thelobster eye.6 In this approach, the focusing device is made ofa stack of flat or curved mirrors and the images given byeach of them are superimposed to form a single image. Theangular acceptance by each mirror should be kept suffi-ciently small to minimize the aberrations. If the mountingerrors of individual mirrors are negligible, sharp images maybe generated as with arefractive lens. The overall angularacceptance of the focusing device is limited only by the criti-cal angle of total external reflection of the mirror, which iswavelength dependent. Later, this idea was further developedin x-ray astronomy and has turned into an imaging technol-ogy known as the lobster-eye microchannel plate optics.7–10

In this article, we describe a general approach for thedesign of a neutron lens based on superposition of glancingreflections followed by a practical design using silicon wa-

fers, a fairly transparent material for neutrons. The design isfurther demonstrated with Monte Carlo simulation for a pro-totype that delivers a 0.1 mm wide neutron beam for a largerange of neutron wavelengths for time-of-flight diffractionapplications.

The proposed lens design is schematically shown in Fig.1. The lens is made of a stack of silicon wafers, each coatedwith a reflective multilayer~supermirror! on one side and aneutron-absorbing layer on the other side~see insert of Fig.1!. All wafers are curved in the horizontal~x–y! plane to afixed radius of curvature,R0, with the reflective multilayeron the convex side of the wafer. Each multilayer will reflectneutrons as a cylindrical concave mirror in a limited angularrange determined by the multilayer length and the nominalsource–mirror distance. Stacks of silicon wafers with coatedreflecting and/or absorbing surfaces have already been usedin neutron scattering instruments as collimators,11,12

benders,13 or multiple reflection focusing microguides.14,15

More recently, Johnson and Daymond suggested the possi-bility of using sets of silicon-based elliptic or parabolic re-flecting surfaces as a neutron lens.16

Unlike the lobster eye imaging optics in x-ray astronomywhere the incoming beams are nearly parallel, imaging apoint neutron source requires a variable aspect ratio~length/thickness! for each mirror to ensure a continuous angularcoverage starting with a minimum incident angle determinedby the maximum allowable aspect ratio. The upper limit ofthe incident angle is defined by the geometry or by the re-flection critical angle. All neutrons whose incident angle fallsin this angular range will be reflected. There are two basicways of varying the channel aspect ratio:~1! changing thethickness while keeping constant length;16 and ~2! changingthe length while keeping constant thickness. We choose touse the second method for practical reasons.

Thus, in the present design, the waferthickness, g0, andthe length, L0, are constant, but thelength of the deposition,Ln ~for the nth mirror! is variable~see Fig. 1!. Only mirrorson one side of the optical axis are considered, and the directbeams are blocked with a neutron absorber. By bending thechannels to a properly chosen curvature radius, the spatialaberration can be significantly suppressed compared to the

a!Electronic mail: stoicaad@ornl.govb!Also at: Metals and Ceramics Division, Oak Ridge National Laboratory,

Oak Ridge, Tennessee 37830.

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 74, NUMBER 4 APRIL 2003

24630034-6748/2003/74(4)/2463/4/$20.00 © 2003 American Institute of Physics

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flat channel case. Assembling properly tapered wafers caneither magnify or demagnify the image. The magnificationratio is given byM5 f 2 / f 1 , wheref 1 and f 2 are the source-to-lens and image-to-lens distances, respectively. We discussonly the demagnifying case, i.e.,M<1, as the magnifyingcase is simply a reverse of the optics.

To maximize the device efficiency, we must eliminatethe possible gaps or overlap between the angular ranges cov-ered by neighboring mirrors. Therefore, a set of recurrentconditions has to be fulfilled between the ends of consecutivemirrors. To facilitate discussions, we denote the mirror num-bern by n5yn /g0 , whereyn is the distance of the mirror tothe optical axis, andn1 andn2 the numbers of the first andthe last mirror, respectively. The solution to the equation setby the recurrent conditions usually oscillates, and the oscil-lation amplifies with the mirror number,n. Fortunately, anapproximate, nonoscillating solution can be obtained with apolynomial trial function of 1/n, which yields

xnL

f 252

2

42CnF11

2~12b!2~3b21!Cn

62Cn3

1

bnG31

bn,

~1!xn

R

f 25

22Cn

42CnF12

2~12b!1~2b21!Cn

62Cn3

1

bnG31

bn,

wherexnL andxn

R are thex coordinates of the left- and right-hand side ends of mirrorn. b5(11M )/2, andCn52b(12 n0 /n), where

n054

~11M !2•

f 22

R0g0~2!

indicates the location of the mirror having the perfect curva-ture for focusing for a given radius of curvature,R0. In theideal case of focusing (Cn50) and to the first order of 1/n,

xnR52xn

L5f 2

2bn. ~3!

This shows that the mirrors are symmetric about they axis,with a total length of

Ln5f 2g0

byn, ~4!

thus giving the lens a 1/y form factor. For uniform bending,CnÞ0 for all n, but numerical solutions forxn

L andxnR from

Eq. ~1! are still well described by Eq.~3! so that the relative

error introduced by the approximate relationship decreasesasymptotically asb/n. Calculations withn251000 andn1

510 indicate an average error of less than 1% depending onM. This gives the confidence that the symmetric solutiongiven by Eq.~3! provides an adequate description of the lensshape in the case of uniform bending. We have chosen to useEq. ~3! in the design due to its simplicity.

The neutron lens design begins by setting the desiredangular acceptance at the sample position, i.e., the lower an-gular limit, gn1, and the upper angular limit,gn2, both mea-sured relative to the optical axis. By rearranging Eq.~4!, weobtain gn5yn / f 2 5 g0 /bLn . Thus, the lower angular limitdetermines the wafer aspect ratio through the relationg0 /L0<bgn1 ~becauseL0>Ln1). By considering a reason-able value forgn1;0.2°, and the fact that the maximumwafer length should lie in the 50–100 mm range for adequateneutron transmission through silicon, we have concluded thatg0 should not exceed 0.35 mm. The upper angular limit is setby the critical anglewc for total reflection,wc'bgn2 . Notethat wc is neutron wavelength dependent.

Another required parameter is the second focusinglength, f 2, which should significantly exceed the radius ofthe ‘‘dead zone’’ occupied by either a large specimen orsample environments. Moreover, it is desirable to separatethe reflected beam from a possible transmitted beam. Theseconsiderations have led us to conclude that the second focus-ing length should be two to three times larger than the deadzone radius.

Knowing the required values ofg0, L0 and f 2, the lensshape is completely set, withn1> f 2 /bL0 and n2

'wcL0 /g0 n1 . The symmetrical lens design prescribes thereflector length asLn5 f 2 /bn . It can be seen that the sameset of mirrors can be used for different degrees of demagni-fication by simply changing the tapering angle, in which casea different focal length will result.

For uniform bending, the image thus formed is not per-fect and exhibits spatial aberrations or point spread. There-fore, the radius of curvature,R0, must be optimized to mini-mize the optical aberrations. Analytical results can bederived for the symmetrical case. The smallest standard de-viation of the point spread results whenn05n1n2 /(n2

2n1) ln(n2 /n1), with s5s0A12 n02/(n1n2), where s0

5bg0 /A3 is the standard deviation of the point spread ofthe lens made of flat mirrors. Thus, the optimalR0 is deter-mined from Eq.~2!.

As a practical example, we consider the neutron lensneeded by VULCAN, a materials science and engineeringdiffractometer under construction at the Spallation NeutronSource.17 One of the design goals for VULCAN is to enablehigh-resolution spatial mapping with a one-dimensional~1D!resolution of ;0.1 mm. The following design parameterswere chosen:M51, f 253 m,L0560 mm, andg050.2 mm.The minimal angle accepted by the lens isgn1'0.2°. Eachchannel reflects neutrons in an average angular range of0.004°. An overall acceptance of 0.31°, adequate for mostdiffraction experiments, is obtained for 80 wafers with a totalthickness of 16 mm. This givesgn2'0.51°. The first and lastmirrors aren1550 andn25130, respectively. The optimalbending radius was determined to beR05584 m. The stan-

FIG. 1. Schematic diagram for the neutron lens withM 5 1 for a pointsource aty050. The vertical scale is highly exaggerated to show details ofthe lens design.

2464 Rev. Sci. Instrum., Vol. 74, No. 4, April 2003 A. D. Stoica and X. L. Wang

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dard deviation of the point spread iss50.03 mm. For aGaussian distribution, this gives a full width at half maxi-mum ~FWHM!52.353s50.07 mm.

To ascertain that the design is correct, a Monte Carlosimulation was carried out using the same set of parameters.Figure 2 shows the normalized image profile calculated bythe Monte Carlo method for a point source at variousy0

positions in the focal plane~see Fig. 1!. While the intensityprofile is almost triangular fory0,0 ~located on the otherside of the line of sight!, long tails with a sharper peak ap-pear for y0.0. However, in all cases, the FWHM of thespatial distribution for the image is less than 0.1 mm. Withsuch a lens, the field of view is at least several mm on eachside of the focus, although the transmission decreases whenmoving away from the optical axis~see next!.

Using the Monte Carlo method, we also estimated thetransmission of the lens for two types of supermirrors,m53.5 andm52. The parameterm denotes the multiplier ofthe critical angle as compared to that of a plain nickel guide,i.e., uc5muNi whereuNi50.1 deg/Å.m53.5 represents thestate of the art whilem52 the economical solution. Forthese calculations, the characteristic reflectivity profile of thesupermirror has to be taken into account and so has the trans-mission through silicon.18 The calculated transmission fory050 is shown in Fig. 3 as a function of neutron wave-length. Even form52, the transmission exceeds 80% forl.2.5 Å. At short wavelengths, the transmission is limited bythe critical angle and thus a quick drop off is seen. Never-theless, the calculations show at least 35% transmission withm53.5, for l.1 Å. Note that the geometric transmissiondecreases if the point source if placed off the optical axis.For y0568 mm, for example, the overall transmission isonly about 50% of that aty050. The reason for the decreasein transmission is quite simple. Fory0.0, gaps appear be-tween the angular ranges covered by consecutive mirrors;some neutrons are transmitted through the device withoutbeing reflected. Fory0,0, some of the neutrons strike theabsorbing layer after reflection.

The device just discussed allows focusing~imaging! in1D. Focusing in two dimensions can be realized viaKirkpatrick–Baez arrangements using two 1D lenses withtheir reflecting surfaces normal to each other. Larger angularacceptance may be obtained with supermirrors that havehigher critical angles. In addition, several lenses could bemounted in parallel to focus neutrons coming from differentsources. In this way, convergent beams with an angular di-vergence of 1°–2° may be delivered onto the sample in spiteof the limitation bywc .

To achieve the performance predicted by the present cal-culations, some technical requirements should be fulfilledconcerning the mirror quality, wafer assembling accuracy,and curvature radius control. These issues are subject of fu-ture investigations.

In summary, we have shown that imaging by superposi-tion of glancing reflections provides a useful means for fo-cusing neutron beams coming from point sources. A practicaldesign was demonstrated for a neutron lens made of a stackof silicon wafers to deliver a 0.1 mm wide neutron beamwith 0.3° angular divergence for time-of-flight diffraction ap-plications. The performance of the lens was checked withMonte Carlo computer simulation. The success of this ex-ample shows that using the basic equations given in thisarticle, the initial design of a neutron lens can be done sim-ply with spreadsheet calculations.

This research was supported by the U.S. Department ofEnergy, Division of Material Science and Engineering. OakRidge National Laboratory is managed by UT-Battelle, LLC,for the US Department of Energy under Contract No. DE-AC05-00R22725.

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FIG. 2. ~Color! Intensity distribution at the image position as function ofsource coordinate,y0. The symbolyiwas used for the coordinate across theimage as shown in Fig. 1. The intensity at a giveny0 has been normalizedby the transmission. Note that the colors are in log scale to illustrate theappearance of long tails fory0.0.

FIG. 3. Neutron transmission as function of wavelength for the neutron lensunder consideration. The solid line shows the transmission through 60 mmof silicon, without considering the reflection on supermirror. The dashed anddotted lines are calculated for two different type of supermirror:m53.5 andm52.

2465Rev. Sci. Instrum., Vol. 74, No. 4, April 2003 Neutron lens by glancing reflections

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2466 Rev. Sci. Instrum., Vol. 74, No. 4, April 2003 A. D. Stoica and X. L. Wang

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